A robust Singular Value Decomposition to detect damage under changingoperating conditions and structural uncertainties
Steve Vanlanduit , Eli Parloo and Patrick GuillaumeVrije Universiteit Brussel (VUB)Department of Mechanical Engineering (WERK)Acoustics and Vibration Research Group (AVRG)Pleinlaan 2, B1050 Brussels, BelgiumSteve.Vanlanduit@vub.ac.bewww.avrg.vub.ac.be
ABSTRACT
During the last decades many techniques were proposed to detect damage based on changes of modal parameters.Unfortunately, the modal parameter changes due to varying operating conditions and structural uncertainties (e.g. dueto interproduct variability) are often much more important, and hence they mask the information on the damage stateof the structure. Therefore, most vibration based damage detection techniques only give good results in wellcontrolledlaboratory conditions. In this article, a technique is proposed to detect damage in structures from measurements takenunder diﬀerent conditions (i.e. diﬀerent operational excitation levels, geometrical uncertainties and surface treatmentsof the structure). The method is based on a Robust Singular Value Decomposition (RSVD) which will be introducedin this article. Using the RSVD the distance of an observation to the subspace spanned by the intact measurementscan be computed. Furthermore, from statistics a threshold can be determined to automatically decide, based on theobservation’s distance to the subspace, if the observation comes from a damaged or intact sample. The proposedRSVD method will be compared with an existing method based on the classical leastsquares (LS) SVD. The damagedetection method will be validated on an aluminium beam with diﬀerent damage scenario’s (a saw cut and a fatiguecrack) which is measured in diﬀerent conditions (diﬀerent beams with small dimensional changes, beams covered withdamping material, and diﬀerent operating levels).
1 INTRODUCTION
During the last decades many techniques were proposed to detect damage based on changes of modal parameters (see the literaturesurvey in
[1]
). A common property of these methods  based on changes in modal parameters  is that they are very sensitive tochanges in: a) the environment, b) the operational conditions and c) structural uncertainties. Recently, a few publications appearedthat aim at removing the inﬂuence of these variations. In
[9]
a technique is proposed to eliminate the inﬂuence of operationalconditions. The technique is based on a Singular Value Decomposition (SVD) of a Frequency Response Function (FRF) matrix
H
= [
H
1
,...,H
N
]
, where
H
1
,...,H
N
are FRFs at
N
conditions. In
[5]
another matrix decomposition (Principal ComponentAnalysis or PCA) of a matrix of estimated features (i.e. resonance frequencies) is used to eliminate the inﬂuence of environmentalchanges. Both authors have shown that matrix decompositions can help in removing the inﬂuence of varying conditions of thestructure/measurement. However, when damaged features (FRFs or estimated parameters) are included in the matrix
H
, thedecomposition largely deviates from the one based on merely features from intact structures. This is true since damaged featuresare outliers, as was recognized by Worden in
[11]
(in the paper
[11]
the fact that a damaged feature is an outlier with respect to thefeatures of intact samples is used to separate damaged and intact observations).
The purpose of this article is to present a robust SVD which computes the SVD of the intact structures from a set of observationsof both intact and damaged structures, which are possibly measured in diﬀerent conditions. In order to guarantee the success of the proposed technique, it is assumed that there are at least as many intact as damaged observations (FRFs, response spectra orestimated parameters) are present in the observation matrix
H
which has to be decomposed.In the next section the theory behind the method will be described. Firstly, in Section 2.1 a review of damage detection using theleastsquares SVD will be given. Next, in Section 2.2, a somewhat more robust iterative SVD (ISVD) is introduced. The proposedrobust SVD (RSVD) is described in Section 2.3. All three methods are validated on an extensive experimental case study of aaluminium beam measured under various conditions. The results of the experimental validation can be found in Section 3. Finallyconclusions will be drawn in Section 4.
2 THEORY2.1 Classical Leastsquares SVD
Assume that
H
= [
H
1
,...,H
N
]
is a matrix of structural observation features taken from
N
experimental conditions (both
N
d
damaged and
N
u
undamaged, where it is assumed that
N
u
≤
N
d
). As will be shown below, the features can be FRFs, responsespectra or estimated modal parameters of the structure. Then, based on the classical leastsquares SVD, an automatic classiﬁcationbetween damaged and intact observations can be made as follows :
Algorithm: Leastsquares SVD based damage detection
1. Compute the SVD of the
M
by
N
matrix
H
:
H
=
USV
H
with
S
=
diag
(
σ
1
,...,σ
min
(
N,M
)
)
.2. Put all singular values
σ
L
+1
,...,σ
min
(
N,M
)
below the noise level equal to zero :
S
1
=
diag
(
σ
1
,...,σ
L
,
0
,...,
0)
.3. Resynthesize the singular vectors to a rank
L
matrix :
H
1
=
US
1
V
H
4. Compute the residual matrix
E
1
:
E
1
=
H
−
H
1
.5. Estimate the standard deviation
s
of the residuals :
s
=
1
MN
−
1
M i
=1
N j
=1
(
H
(
i,j
)
−
H
1
(
i,j
))
2
.6. For each observation
j
(i.e. each column
j
) estimate the standard deviation
s
j
:
s
j
=
1
M
−
1
M i
=1
(
H
(
i,j
)
−
H
1
(
i,j
))
2
.7. Now, if observation
j
(column
j
) can be spanned by the subspace computed using the SVD, the relative distance
d
SVDj
deﬁned by
d
SVDj
=
s
j
s
should  for each
j
 obey a
χ
2
distribution with mean
m
= 1
and standard deviation
σ
=
√
2
M
[3]
.The
α
= 95%
conﬁdence level of the
χ
2
distribution gives the threshold
T
for the distance
d
SVDj
to distinguish between intactand damaged samples : when
d
SVDj
> T
with
T
=
(
M
−
1)
χ
2
100
−
α
2
(
M
−
1)
the structure is damaged (for values
M >
30
amaximal value of
M
= 30
is taken in order to make sure that the threshold is not too thight).For Step 2. of the leastsquares SVD algorithm described above, a subdivision should be made based on the actual feature that isused in the matrix
H
:
•
When FRFs or response spectra (displacements, velocities or outputs) are taken as the columns of
H
, the number of nonzerossingular values (
L
) is equal to
min
(
N
m
,N
o
)
, with
N
m
the number of modes in the frequency band and and
N
o
the numberof measured spectra (outputs) for each observation (this is the case because the rank of an FRF matrix of an
L
DOF systemis equal to
L
).
•
When on the other hand estimated parameters (as for instance the resonance frequencies of the structure) are taken asfeatures, there is only one nonzero singular value. Indeed, for environmental changes or small global structural variabilities(in material properties, surface treatment) resonance frequencies change linearly, and thus all observations can be spannedby one single singular vector (remark that for localized damage on the other hand, a nonlinear shift in resonance frequenciesoccurs).The problem with the leastsquares SVD technique is that it is very sensitive to outliers in the measurements. This is illustrated inFigure 1 for a data matrix
H
=
1
...
161
...
16
, where
H
(16
,
2)
is put equal to 0 to introduce an outlier. The result of the SVDdecomposition in Figure 1 clearly shows that the leastsquares solution is inﬂuenced to an important extent by the single outlier.
0 5 10 150246810121416Dimension 1
D i m e n s i o n 2
Figure 1: Illustration of the eﬀect of an outlier on the leastsquares SVD. ’o’: data without outlier, ’+’: data with outlier, full line:leastsquares ﬁt of the data without outliers, dotted line: leastsquares ﬁt of the data with the outlier.
In the next section a more robust SVD based damage detection method will be introduced to reduce the inﬂuence of outliers (i.e.damaged features).
2.2 Iterative SVD
The ﬁrst approach to reduce the inﬂuence of outliers is to use a twostep iteration approach :
•
Use the classical leastsquares SVD algorithm from the previous section.
•
Eliminate observations with an outlying distance
d
SVDj
.
•
Compute an SVD of the remaining observations.In detail the method contains the following steps :
Iterative SVD (ISVD) based damage detection
1. Steps 1. to 7. are equal to those of the leastsquares SVD presented in the previous section.2. Next, median of the observation distances is computed :
d
med
=
median
(
d
SVD
1
,...,d
SVDN
)
.3. The columns
j
1
,...,j
N
2
which have a distance
d
SVDj
> d
med
are removed from the data matrix :
H
2
=
H
(:
,
{
1
,...,N
}
/
{
j
1
,...,j
N
2
}
)
4. The SVD of the reduced data matrix is computed :
H
2
=
U
2
S
2
V
2
H
with
S
2
=
diag
(
σ
1
,...,σ
min
(
N
2
,M
)
)
.5. Compute the right singular vectors :
V
3
=
U
2
H
S
−
12
H
.6. Resynthesize the singular value decomposition :
H
3
=
U
2
S
3
V
3
H
, with
S
3
(
i,i
) =
S
2
(
i,i
)
for
i
= 1
,...,L
and
S
3
(
i,i
) = 0
for
i > L
.7. Compute the residual matrix
E
3
:
E
3
=
H
−
H
3
.
8. Estimate the standard deviation
s
of the residuals using the Median Absolute Deviation (MAD) as a robust scale estimator
[4]
:
s
=
MAD
(
E
3
)
with
MAD
(
x
) = 1
.
4826
∗
median
(

x
−
median
(
x
)

)
(remark that this is necessary because half of theobservation could be outliers, which would lead to a large increase of the classical sample variance).9. For each observation
j
(i.e. each column
j
) estimate the standard deviation
s
j
:
s
j
=
1
M
−
1
M i
=1
(
H
(
i,j
)
−
H
3
(
i,j
))
2
.10. Now, if observation
j
can be spanned by the
L
dimensional subspace computed using the SVD, the residual for each
j
thedistance
d
ISVDj
=
s
j
s
should obey a
χ
2
distribution with mean
m
= 1
and standard deviation
σ
=
√
2
M
. The
α
= 95%
conﬁdence level of the
χ
2
distribution gives the threshold
T
=
(
M
−
1)
χ
2
100
−
α
2
(
M
−
1)
for the distance
d
ISVDj
to distinguishbetween intact and damaged samples : when
d
ISVDj
> T
the sample corresponding to observation
j
is considered damaged.
2.3 Robust SVD
The Robust SVD method (RSVD) that is proposed in this section is the SVD counterpart of the socalled Least Trimmed Squaresprocedure which was developed for linear regression in
[7]
(later on, a faster variant of the algorithm was implemented in
[8]
).The proposed RSVD method is conceptually very simple :
•
Take a random subset of half of the columns in the data matrix
•
Compute the SVD of these columns
•
Extend the right singular matrix to all the columns in the data matrix.
•
Calculate the residual between the srcinal data matrix and the resynthesized matrix based on the ﬁrst
L
singular vectorscorresponding to the combination of columns.
•
Compute the RMS value of the 50% smallest distances
d
k
1
,...,d
k
N
2
as the cost function
κ
.
•
The SVD of the combination of columns with the smallest cost function
κ
is taken as the RSVD solution.In more detail, the following steps have to be performed :
Robust SVD damage detection
1. Construct a matrix
C
with in the rows all combinations of
N
2
observations (column numbers of the
H
matrix) out of thetotal of
N
observations.2. For each of these
N
!(
N
!
/
2)
2
combinations
i
1
,...,i
N
2
 i.e. rows of
C
 (or
250
randomly selected combinations in case
N >
10
) :(a) Remove columns
i
1
,...,i
N
2
from the data matrix
H
:
H
4
=
H
(:
,
{
1
,...N
}
/
{
i
1
,...,i
N
2
}
)
.(b) Compute the SVD of
H
4
:
H
4
=
U
4
S
4
V
4
H
with
S
4
=
diag
(
σ
1
,...,σ
min
(
N
2
,M
)
)
.(c) Compute the extended right singular vectors :
V
5
=
U
4
H
S
4
−
12
H
.(d) Resynthesize the singular value decomposition :
H
5
=
U
4
S
5
V
5
H
, with
S
5
(
i,i
) =
S
4
(
i,i
)
for
i
= 1
,...,L
and
S
5
(
i,i
) = 0
for
i > L
.(e) Compute the residual
E
5
:
E
5
=
H
−
H
5
.(f) Calculate the distances
d
j
=
s
j
s
(as in the ISVD algorithm).(g) Evaluate the cost function
κ
:
κ
=
N
2
k
=1

d
k
:
N

2
3. The SVD of the combination of columns with the smallest cost function
κ
is taken as the RSVD solution.4. The residual
E
6
=
H
6
−
H
between the data
H
and the resynthesized RSVD
H
6
is used to compute the distances
d
RSVDj
=
s
j
s
where
s
=
MAD
(
E
3
)
and
s
j
=
1
M
−
1
M i
=1
(
H
(
i,j
)
−
H
3
(
i,j
))
2
.5. The test statistic
T
=
(
M
−
1)
χ
2
100
−
α
2
(
M
−
1)
is used to classify the intact and damaged observations :
d
j
> T
for adamaged structure.
3 EXPERIMENTAL RESULTS3.1 Setup
In this paper nine diﬀerent experimental Cases are considered in order to be able to compare the sensitivity of the proposed damagedetection technique (see Section 2) in the presence of structural uncertainties with respect to :
•
the structures’ geometry (product variability)
•
the surface treatment and boundary conditionsThe following cases are tested :
•
Case 1:
Aluminium beam 1 (dimensions
400
x
10
x
5
mm
), intact.
•
Case 2:
Beam from Case 1 covered with plastic tape (see Figure 2a).
•
Case 3:
Beam from Case 1 covered with acoustic damping material (see Figure 2b).
•
Case 4:
Aluminium beam 2 (dimensions as beam 1 with small variability), intact.
•
Case 5:
Aluminium beam 3 (dimensions as beam 1 with small variability), intact.
•
Case 6:
Beam from Case 4 with 30% through saw cut in the middle.
•
Case 7:
Beam from Case 4 with 50% through saw cut in the middle (see Figure 2c).
•
Case 8:
Beam from Case 5 with small fatigue crack.
•
Case 9:
Beam from Case 5 with large fatigue crack (see Figure 2d).Remark that the ﬁrst ﬁve Cases represent an intact realization of the structure, while the last four Cases are damaged. The resultsare given in Section 3.3.Moreover, in order to evaluate the sensitivity to changing operating conditions, each of the Cases 19 was measured using 4 operatingconditions corresponding to excitation levels 0.5V, 1V, 2V and 4V. In Section 3.2 it will be shown that it is possible to eliminate theinﬂuence of the operating conditions in the damage detection process.The beams in the 36 experiments (9 Cases and 4 excitation levels) were supported using thin nylon threads (see Figure 3). Aloudspeaker was used to excite the beams with a periodic chirp (0.5V, 1V, 2V and 4V amplitudes) in the 100Hz4000Hz frequencyrange with a 1.25Hz resolution. A Polytec scanning laser vibrometer was used to measure the response velocity at 25 equidistantlocations of the beams in the diﬀerent experiments.
3.2 Damage detection with changing operating conditions
In ﬁrst instance, the following velocity spectra where used as inputs for the proposed damage detection method (see Figure 4):
•
Case 5 with excitation levels 0.5V,1V,2V and 4V
•
Case 8 with excitation levels 2V and 4V
•
Case 9 with excitation levels 2V and 4V